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Theorem llynlly 23485
Description: A locally 𝐴 space is n-locally 𝐴: the "n-locally" predicate is the weaker notion. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
llynlly (𝐽 ∈ Locally 𝐴𝐽 ∈ 𝑛-Locally 𝐴)

Proof of Theorem llynlly
Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 llytop 23480 . 2 (𝐽 ∈ Locally 𝐴𝐽 ∈ Top)
2 llyi 23482 . . . . 5 ((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) → ∃𝑢𝐽 (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
3 simpl1 1192 . . . . . . . 8 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Locally 𝐴)
43, 1syl 17 . . . . . . 7 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
5 simprl 771 . . . . . . 7 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢𝐽)
6 simprr2 1223 . . . . . . 7 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑦𝑢)
7 opnneip 23127 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑢𝐽𝑦𝑢) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
84, 5, 6, 7syl3anc 1373 . . . . . 6 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
9 simprr1 1222 . . . . . . 7 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢𝑥)
10 velpw 4605 . . . . . . 7 (𝑢 ∈ 𝒫 𝑥𝑢𝑥)
119, 10sylibr 234 . . . . . 6 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢 ∈ 𝒫 𝑥)
128, 11elind 4200 . . . . 5 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥))
13 simprr3 1224 . . . . 5 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝐽t 𝑢) ∈ 𝐴)
142, 12, 13reximssdv 3173 . . . 4 ((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴)
15143expb 1121 . . 3 ((𝐽 ∈ Locally 𝐴 ∧ (𝑥𝐽𝑦𝑥)) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴)
1615ralrimivva 3202 . 2 (𝐽 ∈ Locally 𝐴 → ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴)
17 isnlly 23477 . 2 (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴))
181, 16, 17sylanbrc 583 1 (𝐽 ∈ Locally 𝐴𝐽 ∈ 𝑛-Locally 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087  wcel 2108  wral 3061  wrex 3070  cin 3950  wss 3951  𝒫 cpw 4600  {csn 4626  cfv 6561  (class class class)co 7431  t crest 17465  Topctop 22899  neicnei 23105  Locally clly 23472  𝑛-Locally cnlly 23473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-top 22900  df-nei 23106  df-lly 23474  df-nlly 23475
This theorem is referenced by:  llyssnlly  23486  efmndtmd  24109
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